On Huppert's condition $B$

Let $\mathcal{F}$ be a saturated formation of finite soluble groups. $\mathcal{F}$ is said to satisfy condition B$$ if and only if (a) $\mathcal{F}$ is subgroup-closed, and (b) $G \in \mathcal{F}$ and $N$ a minimal normal subgroup of $G$ implies $\operatorname{Aut} (N) \in \mathfrak{F}$. The purpose of this note is to characterize those saturated formations of finite soluble groups which satisfy condition ${\mathbf{B}}$.